Fast Multiplier for FIR Filters

57
G. Rajesh Babu and C. N. Marimuthu, “Fast multiplier for FIR filters,” International Journal of Scientific and Technical Advancements,
Volume 2, Issue 1, pp. 57-61, 2016.
International Journal of Scientific and Technical Advancements
ISSN: 2454-1532
Fast Multiplier for FIR Filters
G. Rajesh Babu1
, C. N. Marimuthu2
1
Assistant Professor, ECE, SVS College of Engineering, Coimbatore
2
Professor, ECE, Nandha Engineering College, Erode
Email address: rajeshbabu.g@gmail.com
Abstract—RTMCMA (Rounded Truncated Multiple Constant Multiplication and Accumulation) in direct FIR structure is used in FIR filter
design. The proposed technique is n/2 number of partial product achieved via two’s complement method. The Modified Booth Encoding
algorithm is most important algorithm for truncated multiplier and multiplier take important role in high performance system to reduce
power, area and delay optimized is very important in high performance system and DSP system. Truncation error is not more than 1 ulp
(unit of least position). So there is no need of error compensation circuits and the final output will be precised.
Keywords—FIR; booth multiplier.
I. INTRODUCTION
n Microprocessor and DSP high performance systems
uses 70% instructions perform addition and
multiplication. There is need of high speed multiplier.
Low power design directly reduces the operation time in most
of the portable devices. Low power VLSI design has reducing
the power consumption of multiplication algorithms with
having high-speed structures and appropriate performance.
II. REVIEW OF MEMORY BASED ALGORITHMS
Fixed Width Multipliers
Fixed-width design of signed-digit redundant multipliers
that are frequently adopted in high-speed applications. Here
consider the fixed-width multiplier design where the Modified
Booth Encoded (MBE) partial products are represented by
binary signed digits with digit set {+1, 0, +1}. The advantages
of the MBE redundant multiplier are the reduced number of
partial products and the carry-free partial product addition in
the tree structure. Besides, using redundant representations can
avoid the sign extension problem. In order to reduce the
truncation errors and make the error compensation circuit. The
influence of the digits in the minor group can be approximated
by a simple combination of the digits in the major group,
leading to a very simple compensation circuit.
Fast Parallel Multiplier-Accumulator
The parallel multiplier presents a dependence graph (DG)
to visualize and describe the Modified Booth Algorithm
(MBA). Booth multiplier, and the accumulator sections to
ensure the fastest possible implementation. This multiplier
independent of data word size and easy to designing optimum
structures with minimal delay. The pipelined parallel MAC
design is three times faster than other parallel MAC schemes
that are based on the MBA.
Booth Multiplier
Booth multiplication produce the product of two binary
numbers like X and Y, which are having m and n number of
bits(m and n are equal) using 2’s complement representation.
Procedure for implementing the booth algorithm.
1. Making booth table: In booth table there are four
columns, one column for multiplier, second for
previous first LSB of multiplier and other two columns
for partial product accumulator (P). From two
numbers, choose multiplier (X) and multiplicand (Y).
Take 2’s complement of multiplicand (Y). Load X
value in the table. Load 0 for X-1 value. Load 0 in U
and V which will have product of X & Y at the end of
the operation.
2. Booth algorithm requires a detailed inspection of the
multiplier bits for shifting of the partial product (P).
Before shifting, the multiplicand may be added to P,
subtracted from the P, or left unchanged based on the
following rules:1. {Xi Xi-1} is {0 0} Shift only, {Xi
Xi-1} is {1 1} Shift only, {Xi Xi-1} is {0 1} Add Y to
U and shift, {Xi Xi-1} is {1 0} Minus Y from U and
shift.
3. Concatenate U and V and shift arithmetic right shift
which preserves the sign bit of 2’s complement number.
4. Circularly right shift X.
Repeat the same steps until nth bit completed.
III. EXISTING SYSTEM
A parallel tree multiplier designing steps are Partial
Product (PP) generation, PP reduction, and final carry
propagate addition. In deletion operation removes all the
avoidable partial product bits which are shown by the light
gray dots figure 1. Deletion error should be in the range −0.5
ulp ≤ ED≤ 0. From column 3 onwards deletion starts while
first two of partial product bits are kept. After the deletion of
partial product bits, perform column-by-column reduction
method.
When the reduction operation is over, perform the
truncation, which will further remove the first row of (n-1) bits
from column 1 to column (n-1). It will produce the truncation
error which is in the range of −0.5 ulp ≤ ET≤ 0. So the
adjusted truncation error is −0.25 ulp ≤ T≤0.25 ulp. here add a
bias constant of 0.25 ulp for rounding. All the operations
performed finally the PP bits are added to generate final
product. Rounding error is in the form of - 0.5 ulp ≤ ER≤0.5
ulp. The faithfully rounded truncated multiplier, total error is
in range of –ulp< E=(ED+ET+ER) ≤ulp.
I

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Volume 2, Issue 1, pp. 57-61, 2016.
ISSN: 2454-1532
Fig. 1. Deletion, reduction, truncation, and rounding plus final addition.
Fig. 2. Multiplication/Accumulation using Individual and Combined PP
Compression.
Fig. 3. Existing Truncated Multiplier designs.

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Volume 2, Issue 1, pp. 57-61, 2016.
ISSN: 2454-1532
Deletion is performed in Stage 1 to remove the PP bits,
whereas the magnitude of the total deletion error is no more
than 2−P−1. The truncation error is less than 1 ulp, so the
accuracy of the final result is improved.
Parallel multipliers are produce product with the 2n
number of bits and rounded to n number of bits to avoid
growth in word size. Figure 2 shows the difference of
individual multiplications and combined multiplication for A
× B + C × D.
PP Truncation and Compression:
The MCMA module is realized by combining all the partial
products (PPs) where unnecessary PP bits (PPBs) are removed
without affecting the final accuracy outputs. Figure 3 shows
the bit widths of all the filter coefficients are minimized using
non uniform quantization with different word lengths in order
to reduce the hardware cost
This architecture of MCMA with truncation (MCMAT)
that removes unnecessary PPBs. The white circles in the L-
shape block represent the undeletable PPBs. The deletion of
the PPBs is represented by black circles. Figure 4 shows the
simulation results of existing and proposed method and figure
5 shows Power summary of existing method.
Fig. 4. Simulation result of existing truncated multiplier.
The existing truncated multiplication simulation result is
obtained by using ModelSim. The input parameters are forced
for signed multiplication is X=10101010; Y=01010101; and
product result is, Z=00110101; by using rounded truncated
multiplication scheme.
Fig. 5. Power summary of existing method.
Drawback of the truncated multiplier method has more
area, delay and consumes more power.
IV. PROPOSED SYSYTEM - MODIFIED BOOTH MULTIPLIER
Modified Booth (MB) is a prevalent form used in
multiplication. It is a redundant signed-digit radix-4 en-coding
technique. Figure 6 shows block Diagram of Modified Booth
Multiplier.
Drawback of the truncated multiplier method has more
area, delay and consumes more power.
Fig. 6. Block diagram of modified booth multiplier.
Its main advantage is that it reduces by half the number of
partial products in multiplication comparing to any other
radix-2 representation. Let us consider the multiplication of
2’s complement numbers and with each number consisting of
n=2k bits. Figure 7 shows the Add end Generation for
Modified Booth Multiplier.
Fig. 7. Add end generation for modified booth multiplier.

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Volume 2, Issue 1, pp. 57-61, 2016.
ISSN: 2454-1532
One of the solutions of realizing high speed multipliers is
to enhance parallelism which helps to decrease the number of
subsequent calculation stages. The original version of the
Booth algorithm (Radix-2) had two drawbacks.
They are:
The number of add subtract operations and the number of
shift operations becomes variable and becomes inconvenient
in designing parallel multipliers.
The algorithm becomes inefficient when there are isolated
1’s.
These problems are overcome by using modified Radix4
Booth algorithm which scan strings of three bits with the
algorithm given below:
1. Extend the sign bit 1 position if necessary to ensure that
n is even.
2. Append a 0 to the right of the LSB of the multiplier.
3. According to the value of each vector, each Partial
Product will he 0, +y , -y, +2y or -2y. The negative
values of y are made by taking the 2’s complement. The
multiplication of y is done by shifting y by one bit to
the left. Thus, in any case, in designing a n-bit parallel
multipliers, only n/2 partial products are generated.
The modified Booth’s algorithm starts by appending a zero
to the right of X2 (multiplier LSB). Table I shows triplets are
taken beginning at position X-1 and continuing to the MSB
with one bit overlapping between adjacent triplets. If the
number of bits in X (excluding x-1) is odd, the sign (MSB) is
extended one position to ensure that the last triplet contains 3
bits.
TABLE I. Modified Booth Algorithm
Xi Xi-4 Xi-2 Operation Comments
0 0 0 +0 String of zeros
0 1 0 +A A single 1
1 0 0 -2A Beginning of 1’s
1 1 0 -A Beginning of 1’s
0 0 1 +A End of 1’s
0 1 1 +2A End of 1’s
1 0 1 -A A single 0
1 1 1 +0 String of zeros
The low-cost implementations of FIR filters based on the
direct structure in figure with faithfully rounded truncated
multipliers. The MB-RTMCMA module is realized by
accumulating all the partial products (PPs) where unnecessary
PP bits (PPBs) are removed without affecting the final
precision of the outputs.
The bit widths of all the filter coefficients are minimized
using non-uniform quantization with unequal word lengths in
order to reduce the hardware cost while still satisfying the
specification of the frequency response. Figure 8 and 9 shows
the simulation result of FIR filter and proposed truncated
multiplier respectively.
Fig. 8. Simulation result of FIR filter.
Fig. 9. Simulation result of proposed truncated multiplier.
V. EXPERIMENTAL RESULTS AND COMPARISON OF
TRUNCATED MULTIPLIERS
Figure 10 shows the comparison of gate count, power and
delay in bar chart. Table II gives the detailed description of
proposed method is better than existing method.
Fig. 10. Bar chart of existing method and proposed method.
TABLE II. Comparison.
S. No. Description Existing method Proposed method
1 Gate delay 26.51ns 2.63ns
2 Gate count 570 100
3 Power 65mw 55mw

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Volume 2, Issue 1, pp. 57-61, 2016.
ISSN: 2454-1532
VI. CONCLUSION
RTMCMA leads to the smallest area cost, delay and power
consumption. Partial products are generated directly with the
help of booth encoding table. Further booth encoding table is
converted into combination circuits which results into n/2+1
PP rows ,further direct 2’s complement technique are
implement to reduce n/2+1rows to n/2 ,that shows the results
of gate count used for Modified Booth Rounded truncated
multiplier and is less than Rounded truncated multiplier.
Similarly power analysis results show that proposed Modified
Booth Rounded Truncated Multiplier consume less power than
existing Rounded Truncated Multiplier.
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